Nice surprise with the Lambert W. I almost stopped the video when I saw Lambert W, which I usually consider equivalent to giving up on an analytical solution and resorting to a table lookup or numerical solution. Since I can get numerical spreadsheet solutions to these problems without knowing very much, that doesn't normally do anything to extend my analytical skills. In this case, using Lambert W twice to 'undo' itself is very satisfying, and this is the first time I've felt good about its use.
It's clear that there is only one real solution. We know from inspection that x must be positive, since 256^x is always positive. Then, since x•256^x is a strictly increasing function for positive x, it can take on the value 1 at most once. But Lambert's W function is multi-branched, so there is likely an infinite family of complex solutions.
Re- method 1: I smelled Lambert from the get-go, and worked out the correct answer, stopping just short of the X= ... but never saw the 2nd appearance of W() coming! Beautifully done. Mathemagic, indeed!
Given that the W method neant you had to use 4^4 = 256, why not just replace 256 by 4^4 at the outset. Then 4^4x = 1/x 4^4 = (1/x)^(1/x) x = 1/4. Or do method 1 base 4 rather than e.
*sigh* here we go again with overcomplication of a simple problem. Lambert's function does not need to be base e. And once you "magically" guess that 256 is 4^4 then you have 4x*4^4x = 4 and you can simplify to 4x=1 and x=1/4
Another one for the computerphiles? I haven't watched the video but knowing powers of two I am guessing( 1/16 th). I know as easy as 1+1=2 that 16*16=256 :-) ..Now let me see if I am right. And.... egg on the face :-) . I knew it was going to be a power of 2.. I was thinking in the right direction.
@@CriticSimon almost... if the difference is just in numbers it looks like simple TH-cam minutes pumping. lhs is increasing, rhs decreasing. hence just one solution which should obviously be power of 2. 2-3 mins maximum. instead he could at least consider general case: (2^n)^x=1/x for example. and nothing to do with Lambert.
Nice surprise with the Lambert W. I almost stopped the video when I saw Lambert W, which I usually consider equivalent to giving up on an analytical solution and resorting to a table lookup or numerical solution. Since I can get numerical spreadsheet solutions to these problems without knowing very much, that doesn't normally do anything to extend my analytical skills.
In this case, using Lambert W twice to 'undo' itself is very satisfying, and this is the first time I've felt good about its use.
Right on
Right on! First video I have seen that did something with Lambert W more than just conclude W(something) is the answer.
Guessing and checking worked just fine for me.
256^x = 1/x
256 = (1/x)^(1/x)
256 = 2^8 = 2^(2*4) = 4^4
4^4 = (1/x)^(1/x)
1/x = 4
x = 1/4
Using W() twice nearly always means a simpler solution simply matching bases and powers is possible...
Answer x = 1/4
256^ x =1/x
(256^x)^1/x = (1/x)^1/x raised both sides to 1/x
256 =
2^8 = (2^8 =256)
(2^2)^4
4 ^ 4 =(1/x)^1/x
4 = 1/x
x = 1/4 Answer
It's clear that there is only one real solution. We know from inspection that x must be positive, since 256^x is always positive. Then, since x•256^x is a strictly increasing function for positive x, it can take on the value 1 at most once.
But Lambert's W function is multi-branched, so there is likely an infinite family of complex solutions.
256^x=1/x
Raise to 1/x: 256=(1/x)^(1/x)
But 25y=4⁴ --> 4⁴=(1/x)^(1/x)
1/x=4 --> x=¼
👍
Re- method 1: I smelled Lambert from the get-go, and worked out the correct answer, stopping just short of the X= ... but never saw the 2nd appearance of W() coming! Beautifully done. Mathemagic, indeed!
Thank you!
Cannae ye use other branches of W?
Given that the W method neant you had to use 4^4 = 256, why not just replace 256 by 4^4 at the outset.
Then 4^4x = 1/x
4^4 = (1/x)^(1/x)
x = 1/4.
Or do method 1 base 4 rather than e.
similar to what I did
I obtained x=1/4 by applying the laws of exponents and letting u=1/x.
*sigh* here we go again with overcomplication of a simple problem. Lambert's function does not need to be base e. And once you "magically" guess that 256 is 4^4 then you have 4x*4^4x = 4 and you can simplify to 4x=1 and x=1/4
good
4^4=(1/x)^1/x r x= 1/4
Its even so cab be use with - which dosnt work
256=(1/x)^(1/x)
256=2⁸=4⁴=(1/x)^(1/x)
∴1/x=4 →x=1/4
we have to prove it's the only solution
It is obvious
Another one for the computerphiles? I haven't watched the video but knowing powers of two I am guessing( 1/16 th). I know as easy as 1+1=2 that 16*16=256 :-) ..Now let me see if I am right.
And.... egg on the face :-) . I knew it was going to be a power of 2.. I was thinking in the right direction.
Nice!
x = 1/4... 256^(1/4) = ∜256 = √√256 = √16 = 4 = 1/(1/4) = 4/1 = 4. More solutions? No idea!
For all solutions, the approach may be this: 256^x = 1/x
=> x = 256^-x = exp(-x * ln(256)) = exp(-8x * ln(2))
=> x * exp(8x * ln(2) = 1 || * 8 * ln(2)
=> 8x * ln(2) * exp(8x * ln(2)) = 8 * ln(2) || Lambert-W function
=> W(8x * ln(2) * exp(8x * ln(2))) = W(8 * ln(2)) = W(4 * ln(4)) = W(ln(4) * exp(ln(4)))
=> 8x * ln(2) = ln(4) = 2 * ln(2) => x = 2 * ln(2) / (8 * ln(2)) = 1/4. The one I guessed. Only solution within this scope.
X=1/4
x=1/4
I got x=1/4 by inspection.
An Interesting Nonstandard Equation: 256ˣ = 1/x; x =?
256ˣ = (2⁸)ˣ = 1/x = x⁻¹, (x⁻¹)⁻¹⸍ˣ = [(2⁸)ˣ]⁻¹⸍ˣ, x¹⸍ˣ = 2⁻⁸ = (2⁻²)⁴ = (1/4)¹⸍⁽¹⸍⁴⁾; x = 1/4
x*e^(8*x*ln2)=1 , 8*ln2*x*e^(8*ln2*x)=8*ln2 , 8*ln2=4*ln4 , 8*ln2*x*e^(8*ln2*x)=ln4*e^(ln4) , 8*ln2*x=4*ln4 ,
4*ln4*x=ln4 , x=1/4 , test , 256^(1/4)=4 , 1/(1/4)=4 , OK ,
🤮
???
@CriticSimon same old stuff day after day.
@@vladimirkaplun5774 You expect every problem to be of a different type?
@@CriticSimon almost... if the difference is just in numbers it looks like simple TH-cam minutes pumping. lhs is increasing, rhs decreasing. hence just one solution which should obviously be power of 2. 2-3 mins maximum. instead he could at least consider general case: (2^n)^x=1/x for example. and nothing to do with Lambert.
16777216^x=1/x for example
(1/x)^(1/x) = 256 = 2⁸ = 4⁴
1/x = 4 => *x = 1/4*
image search - google
self-deception is a progressive condition
image search - poor judgement
???
x = 1/4